The distribution of Selmer ranks of quadratic twists of Jacobians of hyperelliptic curves

Let $C$ be an odd degree hyperelliptic curve over a number field $K$ and $J$ be its Jacobian. Let $J^{\chi}$ be the quadratic twist of $J$ by a quadratic character $\chi \in \mathrm{Hom} (G_K , \lbrace \pm 1 \rbrace )$. For every non-negative integer $r$, we show the probability that $\mathrm{dim}_{\mathrm{F}_2} (\mathrm{Sel}_2 (J^{\chi} / K)) = r$ for a certain family of quadratic twists can be given explicitly conditional on some heuristic hypothesis.

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Received 31 October 2017

Accepted 22 May 2019

Published 25 October 2019